Question

$\text{Statement 1:}$ The maximum value of $(\sqrt{-3+4x-x^2}+4{)}^2+(x-5{)}^2$ (where $1\le x\le 3)$ is $36.$
$\text{Statement 2:}$ The maximum distance between the points $(5,-4)$ and the point on the circle $(x-2{)}^2+y^2=1$ is $6$

• A. Both the statements are true and $\text{Statement 2}$ is the correct explanation of $\text{Statement 1}$
• B. Both the statements are true but $\text{Statement 2}$ is not the correct explanation of $\text{Statement 1}$
• C. $\text{Statement 1}$ is true and $\text{Statement 2}$ is false
• D. $\text{Statement 1}$ is false and $\text{Statement 2}$ is true

Answer 1

The correct option is A Both the statements are true and $\text{Statement 2}$ is the correct explanation of $\text{Statement 1}$

Let $y=\sqrt{-3+4x-x^2}$
$\Rightarrow x^2+y^2-4x+3=0$ or point $(x,y)$ lies on this circle.

Then, the given expression is $(y+4{)}^2+(x-5{)}^2,$ which is the square of distance between point $P(5,-4)$ and any point on the circle $x^2+y^2-4x+3=0$ which has center $C(2,0)$ and radius $1.$

Now, $CP=5.$
Then the maximum distance between the point $P$ and any point on the circle is $6.$
Therefore, maximum value of $(\sqrt{-3+4x-x^2}+4{)}^2+(x-5{)}^2$ is $36.$